Numerical Methods for Studying Self-similar Propagation of Viscous Gravity Currents
A strongly implicit, nonlinear Crank-Nicolson-based finite-difference scheme was constructed for the numerical study of the self-similar behavior of viscous gravity currents. Viscous gravity currents are low Reynolds number flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. Under the lubrication approximation, the mathematical description of the spreading of these fluids is reduced to solving a nonlinear parabolic partial differential equation for the
... shape of the fluid interface. This thesis focuses on the finite-speed propagation of a power-law non-Newtonian current in a variable width channel-like geometry (a "Hele-Shaw cell") subject to a given mass conservation/balance constraint. The proposed numerical scheme was implemented on a uniform but staggered grid. It is shown to be strongly stable, while possessing formal truncation error that is of second-order in space and it time. The accuracy of the scheme was verified by benchmarking it against established analytical solutions, which were obtained via a first-kind self-similarity transformation. A series of numerical simulations confirmed that the proposed scheme accurately respects the mass conservation/balance constraint. Next, the numerical scheme was used to study the second-kind self-similar behaviour of Newtonian viscous gravity currents flowing towards the end of a converging channel. Second-kind self-similar transformations are not fully specified without further information from simulation or experiment. Thus, using the proposed numerical scheme, the self-similar spreading and leveling leveling of the current was definitively addressed. The numerical results showed favorable comparison with experimental data.